Linear momentum conservation puzzle

For those not familiar with what a typical common-shaft motor-generator set entails: http://www.electrical4u.com/motor-generator-set-m-g-set/
Suppose an initially fully charged battery supplies the motor, a similar but initially depleted battery being charged up by the generator. Each battery adjacent to the respective motor, generator. Energy E thus a mass equivalent E/c^2 ends up being transferred from one location to another. But assuming the entire system is isolated, conservation of momentum requires the centre of mass is not changed by the transfer of mass-energy from battery to battery. At some stage then, one expects an axial force exists along the common shaft (assume a continuous one-piece shaft). Providing just enough impulse to keep the overall centre of mass fixed.

The problem is how, or even if, that is possible given power transmission is via pure torsion in a rotating drive shaft. In more familiar cases; e.g. tandem-shaft belt or chain or gear transmission, or electrical transfer via wires, compensating forces are (relatively) easily identified. Not here.

Looked at the issue many years ago, and iirc it took a good hour or so to arrive at an answer I was reasonably satisfied with. Instead of providing my answer(s) here and now, leaving it as is for ~ 24 hrs. Maybe somewhat more. A teaser challenge. Let's see who here can offer an answer with clear, credible justification, within that rough time frame.

Just knowing that F = dp/dt will or should suggest what specific physical condition in above setup needs focusing on. Actually, there are two convenient limiting cases worth analyzing. The only hint offered is there are just two relevant variables to work with. Angular velocity and torque.
Hand-wavy suggestions won't cut it. An acceptable answer(s) should ideally be quantitative not just qualitative.

If anyone manages to or thinks they have found an essentially direct answer from a textbook or online article etc., be honest enough to admit it and reference to such. Good luck - back later.

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I disagree with your premise that mass- energy is transferred from one battery to the other. The batteries are isolated. The electrons at the power plant are not transferred to my house.

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I think the point is that energy itself confers mass, according to Einstein's mass-energy equivalence relation. A fully charged battery is thus a tiny bit heavier than a discharged one. Nothing to do with more or fewer electrons, it is the electrical potential energy change that is responsible.

So if one battery is used to charge another one, energy flows from one to the other and in consequence the mass of one very slightly decreases while that of the other increases. A tiny amount of mass therefore moves, in effect. Hence the conundrum.

But actually I don't understand why this fancy motor-generator set is introduced. It seems to me the same would occur if a charged 12V battery were connected to a discharged one, with no mechanical moving parts to confuse the issue. I suspect the answer is something to do with an asymmetry in the physical flow of electrons round the circuit which, if charge were transferred from left to right, would lead the system to experience a minute reaction force from right to left. The electrons flowing from left to right would be at a higher potential than those returning from right to left and thus they would presumably be a bit heavier. But I admit this is just my initial feeling about it: I have not thought it through properly - and I am not an expert on relativity problems.

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But actually I don't understand why this fancy motor-generator set is introduced. It seems to me the same would occur if a charged 12V battery were connected to a discharged one, with no mechanical moving parts to confuse the issue. I suspect the answer is something to do with an asymmetry in the physical flow of electrons round the circuit which, if charge were transferred from left to right, would lead the system to experience a minute reaction force from right to left. The electrons flowing from left to right would be at a higher potential than those returning from right to left and thus they would presumably be a bit heavier. But I admit this is just my initial feeling about it: I have not thought it through properly - and I am not an expert on relativity problems.

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If you have electrons flowing from left to right, then they'll exert a force on the battery casing and start it moving right to left, thus conserving momentum. The reason for the fancy motor setup is that no particles of any kind have to cross from one side to the other, so the solution must be more subtle.

If you have electrons flowing from left to right, then they'll exert a force on the battery casing and start it moving right to left, thus conserving momentum. The reason for the fancy motor setup is that no particles of any kind have to cross from one side to the other, so the solution must be more subtle.

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Aha, thank you. Clearly I am limping along some way behind the others on this. I shall think about it and watch to see who else responds.

Replace the batteries with capacitors, is there a center of mass issue?

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The idea of replacing the batteries with capacitors occurred to me as well; the puzzle stays essentially the same. A crucial thing to keep in mind is that the mass issue Q-reeus is taking about has nothing to do with number of electrons. Instead, it comes entirely from the general-relativistic mass/energy equivalence; a set of particles in a high-energy configuration (e.g. a charged battery) will have slightly more mass than the same set of particles in a low-energy configuration (e.g. a discharged battery).

Extrapolating from the capacitor idea, I realized there's an even simpler system that still exhibits the same puzzle while doing away with electronics entirely. Consider a long, thin rod running through the center of a hollow cylinder. Both are highly rigid, although relativity doesn't allow them to be completely rigid. The ends of the rod are attached to the inside of the cylinder by torsion springs, so the tension on the springs varies as the rod rotates. The two springs' equilibrium positions are at different angles of the rod, so as the rod is rotated in either direction from equilibrium, the tension in one spring increases while the tension in the other decreases. If the system starts out of equilibrium and evolves freely, tension will transfer back and forth between the springs, and relativistic mass will do the same.

Using just classical mechanics (or even special relativity), there is no problem; the rod oscillates without any transfer of mass, and the center of mass stays at the geometric center of the system. Under general relativity, some mass-energy moves, so there must be another relativistic effect that counteracts this. I'm not very well-versed in general relativity, so I don't know what that effect might be.

The idea of replacing the batteries with capacitors occurred to me as well; the puzzle stays essentially the same. A crucial thing to keep in mind is that the mass issue Q-reeus is taking about has nothing to do with number of electrons. Instead, it comes entirely from the general-relativistic mass/energy equivalence; a set of particles in a high-energy configuration (e.g. a charged battery) will have slightly more mass than the same set of particles in a low-energy configuration (e.g. a discharged battery).

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Exactly. I had not expected arguments over the specifics in #1, but chime in now to confirm one could have used any power source feeding any type of rotary motor, and any rotary load e.g. friction clutch, flywheel (latter requiring increasing angular speed for an energy transfer to occur.) The key enigma is that mass-energy is transferred from one location to another, along a direction corresponding to the common shaft axis, purely via torsion in that rotating shaft. Show how or if centre of mass remains constant throughout. That's it folks!

Extrapolating from the capacitor idea, I realized there's an even simpler system that still exhibits the same puzzle while doing away with electronics entirely. Consider a long, thin rod running through the center of a hollow cylinder. Both are highly rigid, although relativity doesn't allow them to be completely rigid. The ends of the rod are attached to the inside of the cylinder by torsion springs, so the tension on the springs varies as the rod rotates. The two springs' equilibrium positions are at different angles of the rod, so as the rod is rotated in either direction from equilibrium, the tension in one spring increases while the tension in the other decreases. If the system starts out of equilibrium and evolves freely, tension will transfer back and forth between the springs, and relativistic mass will do the same.

Using just classical mechanics (or even special relativity), there is no problem; the rod oscillates without any transfer of mass, and the center of mass stays at the geometric center of the system. Under general relativity, some mass-energy moves, so there must be another relativistic effect that counteracts this. I'm not very well-versed in general relativity, so I don't know what that effect might be.

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A clever variation! I won't comment on specifics of your analysis for now but please, let's stick to the OP scenario. We can maybe revisit this one later.

A clever variation! I won't comment on specifics of your analysis for now but please, let's stick to the OP scenario. We can maybe revisit this one later.

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Certainly! I only brought up the variation because it highlights what I hope is a useful insight: electrodynamic effects or other subtleties from the EMF coils in the original problem cannot provide the solution.

Not much memory for battery operation, but ions do deposit on electrodes and move. So there is surely a change of mass distribution even in both batteries. Will the positioning of batteries with respect to motor generator make any impact on your question? I think yes.

On depletion of battery power on one side, the motor speed would reduce, so surely the angular momentum of MG set is reducing. There is dP/dt at electrodes, so I think any change will be taken care of at battery level itself. Any change in electron number or motion due to this dP/dt can create some kind of insignificant torque ripple only. So I will go with jerk on battery electrodes only.

Only 2 posters up to having a stab? Both very wrong. Lack of interest? Not going by participation level in the relatively mundane issue contemporary thread here:http://www.sciforums.com/threads/a-problem.158224/
Oh well, can only surmise as to the powerful unmotivator factor(s) here. Best not think out aloud. Will give it some extra time, just in case a light bulb goes on somewhere here in SF land.

Only 2 posters up to having a stab? Both very wrong. Lack of interest? Not going by participation level in the relatively mundane issue contemporary thread here:http://www.sciforums.com/threads/a-problem.158224/
Oh well, can only surmise as to the powerful unmotivator factor(s) here. Best not think out aloud. Will give it some extra time, just in case a light bulb goes on somewhere here in SF land.

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For my part, I haven't taken a stab at a solution because I have little experience with GR, and I know the answer must involve some GR subtleties. If I had to guess, I would hazard that "center of mass" is itself a tricky concept in GR, and if one were to perform the correct sum over 4-vectors, the puzzle would go away. But I have no idea how to do that calculation.

(Also, there is no reason for that other thread to have gone beyond 5 posts at most.)

For my part, I haven't taken a stab at a solution because I have little experience with GR, and I know the answer must involve some GR subtleties. If I had to guess, I would hazard that "center of mass" is itself a tricky concept in GR, and if one were to perform the correct sum over 4-vectors, the puzzle would go away. But I have no idea how to do that calculation.

(Also, there is no reason for that other thread to have gone beyond 5 posts at most.)

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I actually appreciated your input in particular Fednis48. Just trying to stir action from silent bystanders that typically have no hesitation jumping in elsewhere. And quite agree re that other thread.

For those not familiar with what a typical common-shaft motor-generator set entails: http://www.electrical4u.com/motor-generator-set-m-g-set/
Suppose an initially fully charged battery supplies the motor, a similar but initially depleted battery being charged up by the generator. Each battery adjacent to the respective motor, generator. Energy E thus a mass equivalent E/c^2 ends up being transferred from one location to another. But assuming the entire system is isolated, conservation of momentum requires the centre of mass is not changed by the transfer of mass-energy from battery to battery. At some stage then, one expects an axial force exists along the common shaft (assume a continuous one-piece shaft). Providing just enough impulse to keep the overall centre of mass fixed.

The problem is how, or even if, that is possible given power transmission is via pure torsion in a rotating drive shaft. In more familiar cases; e.g. tandem-shaft belt or chain or gear transmission, or electrical transfer via wires, compensating forces are (relatively) easily identified. Not here.

Looked at the issue many years ago, and iirc it took a good hour or so to arrive at an answer I was reasonably satisfied with. Instead of providing my answer(s) here and now, leaving it as is for ~ 24 hrs. Maybe somewhat more. A teaser challenge. Let's see who here can offer an answer with clear, credible justification, within that rough time frame.

Just knowing that F = dp/dt will or should suggest what specific physical condition in above setup needs focusing on. Actually, there are two convenient limiting cases worth analyzing. The only hint offered is there are just two relevant variables to work with. Angular velocity and torque.
Hand-wavy suggestions won't cut it. An acceptable answer(s) should ideally be quantitative not just qualitative.

If anyone manages to or thinks they have found an essentially direct answer from a textbook or online article etc., be honest enough to admit it and reference to such. Good luck - back later.

Click to expand...

Some thoughts:
1). Electrical potential energy reduces in the discharging battery and increases in the charging battery. Therefore, the rest mass of the discharging battery must be decreasing and the rest mass of the charging battery must be increasing according to the principle that associates mass with energy.
2). Electrons do not flow from one battery to the other, so the changes in rest mass have nothing to do with one battery gaining electrons and the other one losing electrons. That would still be true, even if the batteries were directly connected, without the motor/generator between them, because current flows only in a closed loop.
3). Linear momentum is conserved at both ends of the system because there is a complete circuit, complete loop of electron current at both ends. At the discharge battery/motor end, electron current flows from one terminal of the battery, through the motor windings and back to the other terminal of the same battery. This complete loop conserves linear momentum at this end. At the charge generator/battery end, electron current flows from one terminal of the generator output, to one terminal of the battery then (through electro-chemical processes) the current then flows from the other battery terminal back to the generator. This complete loop also conserves linear momentum at this end.
3). With the above considerations, I think the question then boils down to “how is linear momentum conserved across the motor generator”. My answer is energy-mass transportation through the motor/generator does not involve linear momentum or translational kinetic energy; only angular momentum and rotational kinetic energy as well as electromagnetic energy. Both the motor and generator gain mass and energy when they are spinning. I do not think that any axial force along the length of the shaft is required to conserve linear momentum.
4) Mass is conserved; as one battery loses it, the other gains it. I just want to also note that it is the rest mass of the two batteries that is changing, not the so-called “relativistic mass”. Mass is an invariant quantity and the term “relativistic mass” does not apply.

Some thoughts:
1). Electrical potential energy reduces in the discharging battery and increases in the charging battery. Therefore, the rest mass of the discharging battery must be decreasing and the rest mass of the charging battery must be increasing according to the principle that associates mass with energy.
2). Electrons do not flow from one battery to the other, so the changes in rest mass have nothing to do with one battery gaining electrons and the other one losing electrons. That would still be true, even if the batteries were directly connected, without the motor/generator between them, because current flows only in a closed loop.
3). Linear momentum is conserved at both ends of the system because there is a complete circuit, complete loop of electron current at both ends. At the discharge battery/motor end, electron current flows from one terminal of the battery, through the motor windings and back to the other terminal of the same battery. This complete loop conserves linear momentum at this end. At the charge generator/battery end, electron current flows from one terminal of the generator output, to one terminal of the battery then (through electro-chemical processes) the current then flows from the other battery terminal back to the generator. This complete loop also conserves linear momentum at this end.

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So far, so good. To this point, did better than the only other two venturing an answer.

3). With the above considerations, I think the question then boils down to “how is linear momentum conserved across the motor generator”. My answer is energy-mass transportation through the motor/generator does not involve linear momentum or translational kinetic energy; only angular momentum and rotational kinetic energy as well as electromagnetic energy. Both the motor and generator gain mass and energy when they are spinning. I do not think that any axial force along the length of the shaft is required to conserve linear momentum.

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Which postulate contradicts your three points above. Whether you see that or not. Mass decreases at one location and increases by the same amount at another (as a gedanken experiment, ideally lossless energy conversion & transfer is assumed). Without a counteracting linear impulse, directed along the shaft axis, the centre of mass has shifted.

4) Mass is conserved; as one battery loses it, the other gains it. I just want to also note that it is the rest mass of the two batteries that is changing, not the so-called “relativistic mass”. Mass is an invariant quantity and the term “relativistic mass” does not apply.

Which postulate contradicts your three points above. Whether you see that or not. Mass decreases at one location and increases by the same amount at another (as a gedanken experiment, ideally lossless energy conversion & transfer is assumed). Without a counteracting linear impulse, directed along the shaft axis, the centre of mass has shifted.

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Yes, I agree the c.o.m. has shifted.

The shaft is mounted in bearing blocks that exert force to prevent it from moving axially. In such a case I think the linear momentum is not conserved across the drive shaft since the shaft is not a closed system. Only if we include the machine’s frame and the earth is linear momentum conserved.

That is why I only considered the angular momentum for the transfer of mass and angular momentum is conserved.
That is probably wrong, though.

An electric motor providing torque and a generator rotor working on the same rotating shaft is called a dynamotor.

If the dynamotor, the batteries, and the wires connecting them (however long) are considered contributors to the center of mass of the system, then the situation is exactly the same as the center of mass of a battery powering an LED lamp or a light bulb. The electric current results in no net transfer of electron mass or center of mass the system, but it does result in heating the wires interconnecting them, which in turn converts energy stored in the battery into thermal energy radiated into the surroundings in all directions and means of conduction, including the friction of the motor and generator bearings, as well as the heat generated by the chemical energy stored in or discharged from the batteries.

The center of mass situation would be different if the entire system were mounted on wheels mechanically coupled to the same shaft, obviously. If you wanted something to move the center of mass rather than simply rotating the dynamotor shaft or heating its surroundings, this is easily accomplished.